If we are given two Hydrogen atoms, would the only difference between them would be their quantum state (Energy level or eigen value, and the corresponding Orbital or eigen state) and their location (say you are the origin and each of the Hydrogen atoms are located an arm's distance apart)

This leads to my next question which would be what's the difference between two Hydrogen atoms separated arm's distance and a Hydrogen molecule? Wouldn't the only difference be that they are located very close to each other?

2 Answers
2

The difference is that when two hydrogen atoms are up nice and close to one another, their $1s$ orbitals can appreciably mix to generate bonding and antibonding orbitals. The occupied $\sigma$ bonding molecular orbital of $\mathrm{H}_{2}$ (formed by the addition of the $1s$ orbitals) has a negative energy with respect to the $1s$ orbitals of atomic hydrogen. All other molecular orbitals in $\mathrm{H}_{2}$ are virtual. At arm's length, the interaction energy of the two hydrogen atoms is virtually nil.

One talks of a molecule only if the constituents are in a bound state, which implies that they are withnin a microscopic distance of each other.

The probability that the two hydrogen atoms in an $H_2$ molecule are at arm length distance is extremely small, far smaller than inaccuracies of the traditional hydrogen model. Thus it makes no sense to consider it as a real possibility.

Hydrogen atoms are indistinguishable particles, i.e., there is no difference between any two of them. A difference is created only by pointing to one of them, which is then distinguished by its position.